Abstract
In this paper, we find Hall-Shirshov type bases for free pre-Lie algebras. We show that Segal’s basis of a free pre-Lie algebra is a type of these bases. We give a nonassociative Gröbner-Shirshov basis S for a free pre-Lie algebra such that Irr(S) is a monomial basis (called normal words) of a free pre-Lie algebra, where Irr(S) is the set of all nonassociative words, not containing maximal nonassociative words of polynomials from S. We establish the Composition-Diamond lemma for free pre-Lie algebras over the basis of normal words and the degree breadth lexicographic ordering.
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